Denys Dutykh
CNRS–LAMA, University Savoie Mont Blanc, France
Elena Tobisch
arXiv:1411.5518v1 [physics.class-ph] 20 Nov 2014
Johannes Kepler University, Linz, Austria
Resonance enhancement by
suitably chosen frequency
detuning
arXiv.org / hal
RESONANCE ENHANCEMENT BY SUITABLY CHOSEN FREQUENCY
DETUNING
DENYS DUTYKH AND ELENA TOBISCH∗
Abstract. In this Letter we report new effects of resonance detuning on various dynamical parameters of a generic 3-wave system. Namely, for suitably chosen values of
detuning the variation range of amplitudes can be significantly wider than for exact resonance. Moreover, the range of energy variation is not symmetric with respect to the
sign of the detuning. Finally, the period of the energy oscillation exhibits non-monotonic
dependency on the magnitude of detuning. These results have important theoretical implications where nonlinear resonance analysis is involved, such as geophysics, plasma physics,
fluid dynamics. Numerous practical applications are envisageable e.g. in energy harvesting
systems.
Key words and phrases: nonlinear resonance; frequency detuning; 3-wave system;
resonance enhancement.
MSC: [2010]37N10 (primary), 37N05, 76B65 (secondary)
PACS: 05.45.-a, 92.10.hf
∗
Corresponding author.
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2
Model equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3
Amplitudes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
4
Phase space analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
5
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
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D. Dutykh & E. Tobisch
1. Introduction
Numerous natural phenomena exhibit linear and nonlinear resonances. In many technical cases occurrence of resonance must to be avoided, the widely known Tacoma Bridge
dramatic collapse being an example for this. In other cases, the goal is to approach the
state of exact resonance, by reducing resonance detuning, in order to increase the efficiency
of a process or device. To give a notion of linear resonance, we consider a linear oscillator
(or pendulum) driven by a small force. We say, that the resonance occurs, if the eigenfrequency ω of a system coincides with the frequency of the driving force Ω. In this case, for
small enough resonance detuning, ∣Ω−ω∣ > 0, the amplitude of the linear oscillator becomes
smaller with increasing detuning.
The simplest case of nonlinear resonance is a set of 3 waves Aj ei(kj xj −ωj t) fulfilling resonance conditions of the following form
ω1 ± ω2 ± ω3 = 0,
(1.1)
k1 ± k2 ± k3 = 0,
(1.2)
where kj ∈ Z2 , ωj = ω(kj ) are the wave vectors and frequencies respectively. Drawn from
these resonance conditions, resonance detuning in the nonlinear case can be defined in a
number of ways, e.g. as a frequency detuning or phase detuning.
In the physical literature, for frequency detuning defined as
̃ ≪ min {ωj },
ω1 + ω2 − ω3 = ∆ω
j=1,2,3
the assumption prevails, that bigger detuning results in smaller variation of amplitudes,
e.g. [2]. Quite consistently the study of phase detuning in (1.2) in a 3-wave system demonstrated that the variation range of wave amplitudes becomes smaller with growing dynamical phase [1].
In this Letter we study the effects of frequency detuning in (1.1) by means of the numerical simulation. As an example we take a resonant triad of atmospheric planetary
waves from [5]. Our numerical simulations show that the detuned system behaviour was
̃ ≫ δ > 0. However, there exists a transitional range of the
correctly understood for ∣∆ω∣
̃
detuning magnitude ∆ω where the dynamics of this system is much more complicated. As
our results depend only on the general form of the dynamical system given below, they are
applicable to a wide class of physical systems.
2. Model equations
The dynamical system for detuned resonance of three complex-valued amplitudes Ai ,
i = 1, 2, 3 reads
N1 Ȧ1 = −2iZ(N2 − N3 )A∗2 A3 e−i∆ωT ,
N2 Ȧ2 = −2iZ(N3 − N1 )A∗1 A3 e−i∆ωT ,
N3 Ȧ3 = 2iZ(N1 − N2 )A1 A2 ei∆ωT ,
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Resonance enhancement by detuning
Parameter
Value
Resonant wavevectors, [mj , nj ]
[4, 12], [5, 14], [9, 13]
Resonant frequencies, 2mj /nj (nj + 1) 0.0513, 0.0476, 0.0989
156, 210, 182
Resonant triad parameters, Nj
Interaction coefficient, Z
7.82
20%, 30%, 50%
Initial energy distribution (a), %
Initial energy distribution (b), %
40%, 40%, 20%
Initial dynamical phase, ψ
0.0
Table 1. Physical parameters used in numerical simulations.
(and their complex conjugate equations) where the dot denotes differentiation with respect
̃
to the slow time T = t/ε, ∆ω ∶= ∆ω/ε,
ε being a small parameter. The dynamical system
for exact resonance is obtained by setting ∆ω ≡ 0. It can be rewritten in amplitude/phase
variables as:
N1 Ċ1 = −2Z(N2 − N3 )C2 C3 sin ψ,
N2 Ċ2 = −2Z(N3 − N1 )C1 C3 sin ψ,
N3 Ċ3 = −2Z(N1 − N2 )C1 C2 sin ψ,
N2 − N3 −2 N3 − N1 −2 N1 − N2 −2
C1 +
C2 +
C3 ) cos ψ,
ψ̇ = ∆ω − 2ZC1 C2 C3 (
N1
N2
N3
(2.1)
(2.2)
(2.3)
(2.4)
where Ci (T ) = ∣Ai (T )∣ is the real amplitude, ψ ∶= θ1 + θ2 − θ3 is the dynamical phase and
θi (T ) = arg Ai (T ). In what follows we will focus on the evolution of the energy of the
high-frequency mode E3 (T ).
3. Amplitudes
In Fig. 1 we show the energy evolution in the resonant triad given in Table 1 for several
̃ ∈ [− 1 , 1 ]; e.g. in geophysical applications
values of the frequency detuning ∆ω = ∆ω/ε
2 2
ε ∼ O(10−2 ). From these graphs it can be seen that the period τ and the range of the
energy variation, defined as
1
∆E (∆ω) ∶= (max E − min E ),
(3.1)
t
2 t
are non-monotonic functions of the detuning ∆ω.
A graph showing the characteristics of the dependency of the energy variation range
∆E (∆ω) on the frequency detuning ∆ω is shown in Fig. 2. This particular curve was
computed for parameters given in Tab. 1. The graph can conveniently be divided into
the five regions which are separated by particular values of the frequency detuning ∆ω:
(1,2)
∆ωmax correspond to local maxima, ∆ωst is the position of the local minimum, and ∆ω = 0
corresponds to exact resonance. So, the regions are:
(I): ∆ω ∈ (−∞, ∆ωmax ];
(1)
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D. Dutykh & E. Tobisch
Region/Range
(I)
(II)
(III)
(IV)
(V)
Ð→
∆E +, τ +,
∆E −, τ +,
∆E −, τ +,
∆E +, τ −,
∆E −, τ −,
∆ψ−
∆ψ−
∆ψ−
∆ψ+
∆ψ+
←Ð
∆E −, τ −,
∆E +, τ −,
∆E +, τ −,
∆E −, τ −,
∆E +, τ +,
∆ψ+
∆ψ+
∆ψ+
∆ψ−
∆ψ−
Table 2. Behaviour of physical parameters ∆E , τ and ψ in different regions.
(II): ∆ω ∈ (∆ωmax , min{0, ∆ωst }];
(III): ∆ω ∈ (min{0, ∆ωst }, max{0, ∆ωst }];
(2)
(IV): ∆ω ∈ (∆ωst , ∆ωmax ];
(2)
(V): ∆ω ∈ (∆ωmax , +∞).
(1)
The reason to regard these regions separately is that the main characteristics (i.e. energy
variation ∆E , energy oscillation period τ and the phase variation ∆ψ) behave differently in
each region. Our findings are summarized in Table 2, where all the quantities E , τ and ψ
are followed by ± sign denoting the their variation in the region (+: increase, −: decrease).
The first column corresponds to the direction of increasing values of ∆ω ∈ (−∞ → +∞),
while the second column corresponds to the opposite direction ∆ω ∈ (−∞ ← +∞).
The most interesting observation is, that the energy variation range during the system
evolution can be significantly larger for a suitable choice of the detuning ∆ω ≠ 0 compared
to the exact resonance case ∆ω = 0. There are two values of which provide significant
amplifications to ∆E . On Fig. 2 the global maximum is located on the left of ∆ωst , while
on Fig. 3 it is on the right of ∆ωst . These two cases differ only by the initial energy
distribution among the triad modes (see Tab. 1, initial conditions (a) & (b)).
Similar computations have been performed for other resonant triads and the qualitative
behavior of the energy variation has always been similar to Figs. 2 & 3. Namely, the global
maximum is located on the left of ∆ωst when the high frequency mode ω3 contains initially
most of the energy, and to the right of ∆ωst in the opposite case.
It is important to stress that the energy variation ∆E (∆ω) at the global maximum
(g)
(g)
∆ωmax is always significantly higher than at the point of exact resonance, i.e. ∆E (∆ωmax ) >
∆E (0). The highest ratio ∆E (∆ω)/∆E (0) is attained when the local minimum ∆ωst
coincides with the point of exact resonance. In this case we can find a ∆ω that the
amplification ∆E (∆ω)/∆E (0) is of at least one order of magnitude. A simple phase space
analysis allows to locate the local minimum.
4. Phase space analysis
On Figs. 4 – 6 we depict the typical phase portraits of the dynamical system (2.1) – (2.4)
in phase-amplitude variables. For illustration we choose the triad given in Tab. 1 with the
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Resonance enhancement by detuning
∆ω = 0.50
1
Ei (T )
E1(T)
E2(T)
0.5
0
E3(T)
0
50
0
50
Ei (T )
150
100
150
100
150
100
150
∆ω = 0.0
1
Ei (T )
100
0.5
0
0.5
0
0
50
∆ω = −0.25
1
Ei (T )
150
∆ω = 0.25
1
0.5
0
0
50
∆ω = −0.50
1
Ei (T )
100
0.5
0
0
50
T
Figure 1. Energy evolution in the triad given in Table 1, for different values of
the detuning ∆ω.
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D. Dutykh & E. Tobisch
Energy variation range
0.25
1
2 (max{E} − min{E})
0.2
0.15
0.1
0.05
II
I
III
0
−0.5
IV
V
0
0.5
1
∆ω
Figure 2. Typical dependency of the energy variation range ∆E on the frequency
detuning ∆ω for the case when the high frequency mode ω3 has the maximal
energy (initial condition (a)). The vertical red dashed line shows the location of
(1)
(2)
∆ωmax , while the vertical black solid line shows the location of ∆ωmax . Finally,
the blue dash-dotted line shows the amplitude obtained the exact resonance.
Energy variation range
0.4
0.35
0.25
0.2
0.15
1
2
!
t
t
"
max {E} − min{E}
0.3
0.1
0.05
0
−1.5
−1
−0.5
0
0.5
1
1.5
2
∆ω
Figure 3. Typical dependence of the energy variation range ∆E on the
frequency detuning ∆ω for the case when the high frequency mode ω3 has the
lowest energy (initial condition (b)).
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Resonance enhancement by detuning
initial energy distribution (a). In these pictures we represent the high-frequency mode C3
on the horizontal axis, while the dynamical phase ψ is on the vertical.
The main finding is a pronounced asymmetry between the phase portraits for positive
and negative values of the detuning ∆ω: the position of the stationary point right on the
horizontal axis C3 (there are other stationary points for ψ ≠ 0) is very different; the shape
of the periodic orbit differs (see Figs. 4(b) & 5(b)); the transition from closed to snake-like
integral curves takes place for different values of ∣∆ω∣, e.g. ≈ 0.31 on Fig. 4(c) and ≈ −0.24
on Fig. 5(c); the shape of the integral curves is different; the phase portraits look alike, but
differ in size by one order of magnitude. In order to demonstrate how big this difference is
for opposite values of ∆ω we depicted on the same Fig. 6 the periodic cycles from Figs. 4(a)
& 5(a).
A simple phase space analysis reveals the reason for the presence of a local minimum
of ∆E in Figs. 2 & 3. Indeed, it can happen that the initial conditions coincide with the
system equilibrium point, which depends on ∆ωst .
5. Conclusions
It was demonstrated that the introduction of frequency detuning significantly enriches
the dynamics of a 3-wave resonance system. Moreover, the effects of detuning are highly
nonlinear and highly non-monotonic with respect to the detuning parameter. The main
findings of this study are outlined hereinbelow:
● The range of values of frequency detuning ∆ω can most conveniently be divided
into five regions of different behaviour, not all of them present in any case. The
behavior of the main parameters of system (2.1) – (2.4) over those five regions is
summarized in Tab. 2.
● The amplitude of energy variation (3.1) in a triad with suitably chosen detuning
(∆ω ≠ 0) can be significantly higher than in the case of exact resonance, i.e. ∆ω ≡ 0.
The maximal amplification as compared to exact resonance is attained when ∆ωst
coincides with the point of exact resonance. In this case one of the zones (III) or
(IV) disappears.
● The phase portraits along with the shape and size of the periodic cycles are substantially different for ∆ω > 0 and ∆ω < 0. This means that any complete analysis of
detuned resonance must include both positive and negative values of the detuning
parameter ∆ω.
The notion of resonance enhancement via frequency detuning, as the title of this Letter
says, contradicts what we would expect from our physical intuition. However, there exists a
simple qualitative explanation for that. Indeed, our intuition comes from a linear pendulum
ẍ + ω 2 x = 0
(5.1)
taken usually as a model for a linear wave, and a resonance is regarded due to an action
of an external force.
10 / 13
D. Dutykh & E. Tobisch
(a) ∆ω = 0.1
(b) ∆ω = 0.31
(c) ∆ω = 0.315
Figure 4. Phase portraits of the dynamical system (2.1) – (2.4) in (C3 , ψ)
variables for the triad from the Tab. 1(a). Positive increasing detuning.
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Resonance enhancement by detuning
(a) ∆ω = −0.1
(b) ∆ω = −0.235
(c) ∆ω = −0.25
Figure 5. Phase portraits of the dynamical system (2.1) – (2.4) in (C3 , ψ)
variables for the triad from the Tab. 1(a). Negative decreasing detuning.
12 / 13
D. Dutykh & E. Tobisch
Figure 6. Simultaneous plot of two phase portraits and integral curves for the
detunings ∆ω = ±0.1 shown at Figs. 4(a) & 5(a) correspondingly.
On the other hand, the dynamical system for 3-wave resonance can be transformed into
the Mathieu equation which describes a particular case of the motion of an elastic pendulum
2
− λ cos(ωspr )]x = 0,
ẍ + [ωpen
(5.2)
where ωpen and ωstr are frequencies of pendulum- and spring-like motions, [4]. Regarding
̃ as a frequency of an external force for (5.2), our findings can be
resonance detuning ∆ω
understood in the following way. The detuned 3-wave system has the maximal range of
the energy amplitudes variation when the elastic pendulum interacts resonantly with the
external forcing. Detailed study of this effect is upcoming, using the approach developed
in [3] for an elastic pendulum subject to the external force.
Acknowledgments
This research has been supported by the Austrian Science Foundation (FWF) under
projects P22943-N18 and P24671.
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Resonance enhancement by detuning
References
[1] M. D. Bustamante and E. Kartashova. Effect of the dynamical phases on the nonlinear amplitudes’
evolution. EPL, 85(3):34002, Feb. 2009. 4
[2] A. D. D. Craik. Wave Interactions and Fluid Flows. Cambridge University Press, Cambridge, 1988. 4
[3] M. Gitterman. Spring pendulum: Parametric excitation vs an external force. Physica A, 389(16):3101–
3108, Aug. 2010. 12
[4] E. Kartashova. Nonlinear Resonance Analysis. Cambridge University Press, Cambridge, 2010. 12
[5] E. Kartashova and V. S. Lvov. Model of Intraseasonal Oscillations in Earth’s Atmosphere. Phys. Rev.
Lett., 98(19):198501, May 2007. 4
LAMA, UMR 5127 CNRS, Université Savoie Mont Blanc, Campus Scientifique, 73376 Le
Bourget-du-Lac Cedex, France
E-mail address: Denys.Dutykh@univ-savoie.fr
URL: http://www.denys-dutykh.com/
Institute for Analysis, Johannes Kepler University, Linz, Austria
E-mail address: Elena.Tobisch@jku.at
URL: http://www.dynamics-approx.jku.at/lena/